# How does pH depend on temperature, when taking into account volume expansion?

Since autodissociation of water is endothermic, increasing the temperature will drive the following equilibrium to the right, which has the effect of increasing $$[\ce{H+}]$$:

$$\ce{H2O <=> H+ + OH-}$$

However, increasing the temperature of liquid water will also increase its volume, which has the effect of reducing $$[\ce{H+}]$$ (cf. this table on Wikipedia showing how density of water varies with temperature).

So as we increase the temperature, which of the two effects dominates? Does $$[\ce{H+}]$$ experience a net increase or decrease?

• pH is NOT inversely proportional to [H+](it would be pH = k/[H+]), as pH = -log([H+]). Commented May 29, 2020 at 3:33
• The concentration of [H+] is $c\alpha$. The alpha is related to equilibrium constant of dissociation of acid($K_a$). We know that $K_a$ is related to temperature by van't Hoff equation. So $K_a$ increases with temperature if reaction is endothermic , thus increases [H+] and decreasing pH.
– Manu
Commented May 29, 2020 at 6:31
• I edited your question for clarity. Commented May 30, 2020 at 1:13
• Commented May 31, 2020 at 12:21

Your qualitative understanding is correct. As you increase the temperature, you have two effects:

1. The dissociation reaction, which is endothermic, shifts to the right. This effect increases the concentration of hydrogen ions.

2. The concentration of hydrogen ions is expressed as molarity, which is number per unit volume. As you increase $$T$$, water expands, resulting (everything else being equal) in all concentraitons decreasing.

And you're asking which one wins.

The answer is the first, overwhelmingly. For instance, as you go from $$\pu{25 ^\circ C}$$ to $$\pu{50 ^\circ C}$$, $$K_\mathrm w$$ increases by five times. By contrast, the increase in volume over this temperature range is only $$1\%$$.

For pure water, $$\mathrm{pH}=- \frac 12 \cdot \log{K_\mathrm{w}}$$. As $$K_\mathrm{w}$$ grows with temperature, $$\mathrm{pH}$$ decreases.

The influence of water thermal dilation is much smaller than the temperature dependency of water autodissociation constant.

For solutions with acids, bases, acidic or alkaline salts, buffering systems it is much more complicated. It depends how acidity/alkality constants depend on temperature.